7. Geometry

Lesson

We have now seen how to find the volumes of rectangular prisms. We saw that we can think about the area of their base multiplied by their heights or we can just use a formula.

Formulas for volume

In general: |
$\text{Volume }=\text{Area of base }\times\text{Height }$Volume =Area of base ×Height |

Rectangular prism: |
$V=lwh$ |

We will now look at some problem-solving questions with volume. Problem-solving may involve a question with an application, such as filling a pool and calculating the amount of water required. It could also be a question which is just a bit different from the usual scenario, meaning we won't be able to use the formula as is.

Practice questions

A tank has a length of $7$7 m, width of $2$2 m and depth of $6$6 m.

Find the volume of the tank in m

^{3}.

This wild animal house is made out of plywood.

If the nesting box needs to have a volume of $129978$129978 cm^{3} and a height of $83$83 cm and front width of $54$54 cm, find the depth of the box.

We have now seen how to find the surface area of rectangular prisms. We saw that we can think about their nets and add up the areas of all of the faces or use a formula.

Formulas for surface area

In general: |
$\text{S.A. }=\text{sum of areas of faces }$S.A. =sum of areas of faces |

Rectangular prism: |
$\text{S.A. }=2lw+2lh+2wh$S.A. =2 |

We will now look at some problem-solving questions with surface area. Problem-solving may involve a question with an application, such as wrapping a present and calculating the amount of paper required. It could also be a question which is just a little bit different from the usual scenario, which means we won't be able to use the formula as is.

We are told that a cube desk weight has a surface area of $150$150 in^{2}. What is the side length of this cube?

**Think:** A cube has $6$6 identical faces, so each face will have an area of $\frac{1}{6}$16 of the total surface area. We also know that the area of a square is $A=l^2$`A`=`l`2, so we can use that to find the length.

**Do:** This question requires two calculations.

Find the area of one square face:

S.A. of cube | $=$= | $150$150 |
Given |

Area of one face | $=$= | $150\div6$150÷6 |
Dividing by $6$6 for $6$6 identical faces |

$=$= | $25$25 |
Simplifying |

We now know that the area of one square face is $25$25 in^{2}, so we can find the length of the side or edge. We are looking for a number that when multiplied by itself gives $25$25.

We can use the square root or our perfect squares from memory.

$l$l |
$=$= | $\sqrt{25}$√25 |

$l$l |
$=$= | $5$5 |

**Reflect:** Does this answer seem reasonable?

Practice questions

A birthday gift is placed inside the box shown.

What is the minimum amount of wrapping paper needed to wrap this gift? (Assume the box is in the shape of a rectangular prism)

A paint roller is cylindrical in shape. It has a diameter of $6.8$6.8cm and a width of $31.2$31.2cm. Find the area painted by the roller when it makes one revolution, correct to the nearest $0.01$0.01 cm^{2}.

Solve real-world and mathematical problems involving area of two-dimensional objects composed of triangles and trapezoids. Solve surface area problems involving right prisms and right pyramids composed of triangles and trapezoids. Find the volume of right triangular prisms, and solve volume problems involving three- dimensional objects composed of right rectangular prisms.